Subgroups — #63559

Order 288.

119 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic
2	C₃×#208	C₂	1	2	yes	maximal, complemented
3	#823	—	3	2	no	maximal
4	C₂²×C₃⋊D₃	C₄	1	2	yes	—
5	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
6	C₄⋊(C₃⋊D₃)	—	2	2	no	—
7	C₃×C₃⋊C₈	—	2	2	no	—
8	C₂×D₁₂	—	3	2	no	—
9	C₂×D₁₂	—	6	1	no	—
10	C₃×M₁₆	—	3	2	no	—
11	#208	D₃	1	2	yes	complemented
12	C₂²×C₃²	C₂×C₄	1	1	yes	characteristic
13	C₂×C₃⋊D₃	—	2	2	no	—
14	C₂×C₃⋊D₃	—	4	2	no	—
15	C₃²×C₄	D₄	1	2	yes	—
16	#100	—	9	1	no	Sylow
17	C₂²×D₃	—	3	4	no	—
18	C₂²×D₃	—	6	2	no	—
19	C₂×C₃×C₄	—	2	1	no	—
20	C₂×C₃×C₄	C₂×D₃	1	2	yes	—
21	D₁₂	—	6	4	no	—
22	D₁₂	—	6	2	no	—
23	D₁₂	—	6	2	no	—
24	C₃×C₈	—	6	2	no	—
25	C₃⋊C₈	—	2	2	no	—
26	C₂×C₃²	—	2	1	no	—
27	C₂×C₃²	C₂²⋊C₄	1	1	yes	characteristic
28	C₃⋊D₃	—	4	2	no	—
29	C₂×D₄	—	9	1	no	—
30	M₁₆	—	3	2	no	—
31	C₂²×C₃	—	2	1	no	—
32	C₂²×C₃	C₄×D₃	1	2	yes	—
33	C₂×D₃	—	6	4	no	—
34	C₂×D₃	—	6	4	no	—
35	C₂×D₃	—	12	4	no	—
36	C₂×D₃	—	12	4	no	—
37	C₃×C₄	—	2	2	no	—
38	C₃×C₄	D₁₂	1	2	yes	—
39	C₃×C₄	#67	1	2	yes	—
40	C₃²	#100	1	1	yes	characteristic, complemented, Sylow
41	C₂³	—	9	2	no	—
42	D₄	—	18	2	no	—
43	C₂×C₄	D₃²	1	1	yes	characteristic
44	C₈	—	6	2	no	—
45	C₂×C₃	—	2	2	no	—
46	C₂×C₃	—	2	2	no	—
47	C₂×C₃	—	2	1	no	—
48	C₂×C₃	(C₂×D₃)⋊C₄	1	2	yes	—
49	D₃	—	12	4	no	—
50	D₃	—	12	4	no	—
51	C₂²	—	18	2	no	—
52	C₂²	—	36	2	no	—
53	C₂²	#624	1	1	yes	characteristic
54	C₄	#626	1	2	yes	—
55	C₃	—	2	1	no	—
56	C₃	#823	1	2	yes	complemented
57	C₂	—	2	1	no	—
58	C₂	—	36	2	no	—
59	C₂	#3722	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#3722
commutator	C₂²×C₃²	C₂×C₄
Frattini, hypercenter	C₂×C₄	D₃²
Fitting, cosocle	C₂×C₃²×C₄	C₂²
socle	C₂×C₃²	C₂²⋊C₄
prime radical	C₂×C₄⋊(C₃⋊D₃)	C₂
hypocenter	C₃²	#100
LCS	G ▷ C₂²×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₃×#208	C₂	semidirect
11. #208	D₃	semidirect
40. C₃²	#100	semidirect
56. C₃	#823	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄⋊(C₃⋊D₃)	2
C₃×#208	2
#823	3